Plasmonic Waveguides : Design and Comparative Study

Plasmonic Waveguides : Design and Comparative Study

YASSIN CHOWDHURY

Master’s Degree Project Stockholm, Sweden 2011

TRITA-ICT-EX-2011:139

Academic Year 2010-2011

Plasmonic Waveguides : Design and Comparative Study

Yassin Chowdhury

Erasmus Mundus Master in Photonics Master Thesis

Promotor: Prof. Min Qiu

Supervisor: Dr. Min Yan

Abstract

Although photonics offers an attractive solution to the speed limitation of electronics, reducing the size of bulky photonic components is one of the major issues towards the implementation of photonic integrated circuits. Plasmonic circuits, which tightly confine electromagnetic waves at the metal-dielectric interface, can be a potential solution to this problem. Despite an excessive amount of papers published in the field, there exist an inconsistency in terms of the measure of plasmonic modal properties, especially their mode size. In this thesis work, several representative plasmonic structures are studied and there modal characteristics are cross- compared. Confinement has been calculated using several definitions in order to measure their figure of merits consistently. In addition, we propose a plasmonic waveguide, which achieves both deep sub-wavelength scale confinement and relatively long propagation. The waveguide consists a metal nanowire on top of a high index dielectric strip which can be made of Germanium. The hybridization of dielectric mode and cylinder plasmon polariton mode leads to localization of the mode energy in the nanoscale gap. This structure outperforms existing plasmonic waveguides in terms of figure of merit, which makes the waveguide particularly useful in high density photonic integrated circuits. This thesis also investigates optical forces due to the enhanced gradient field in plasmonic waveguides. Gradient forces and trapping forces are calculated using Maxwell’s stress tensor for different hybrid waveguides, among which the forces with the proposed waveguide structure are found to be the largest.

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Acknowledgements

Firstly, I would thank to Allah, for having made everything possible, by giving me the strength, patience and courage to do this work.

This thesis would not have been accomplished without the guidance and support of several individuals during the preparation and completion of the study. I would like to thank my su- pervisor Dr. Min Yan for his abundant help, unselfishness, guidance and prolific suggestions.

Also my promoter Prof. Min Qiu had been the mentor behind the work and I thank him for his encouragement. Thanks goes to all the members of the nanophotonics group and the people of FMI in KTH for the fantastic memories.

I am indebted to the Erasmus Mundus program from the European Commission for the schol- arship I obtained for the educational journey to KTH, Sweden.

My deep gratitude to Prof. Roel Baets for the support that I experienced in this adventure in Photonics for the last two years. I also thank Mr. Bert Coryn, the programme officer of Erasmus Mundus Photonics, for his continuous help and coordination during this program.

My fellow colleagues and friends in Erasmus Mundus - Mesut, Glenn, Joanne, Akin, Imran, Eliyas, Ashim - you guys made this journey so wonderful.

I am deeply thankful to my parents and sisters for unremittingly supporting and encouraging me. Although I am living thousand miles away from home, their prayer and love is always with me.

.. and Iti, my loving wife, without your love, support and understanding, this work would not be possible. I am so thankful that I have you with me, pushing and encouraging me, when I was ready to give up.

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Contents

1 Introduction 1

2 Principle of surface plasmon waveguiding 3

2.1 Introduction . . . . 3

2.2 The electromagnetic wave equation . . . . 4

2.3 Propagating and evanescent waves . . . . 6

2.4 Diffraction limit and low dimensional optical waves . . . . 8

2.4.1 Diffraction limit in 3D waves . . . . 8

2.4.2 Diffraction limit in 2D waves . . . . 9

2.5 Surface plasmon polariton . . . . 10

2.5.1 Dispersion relation of SPP at a single interface . . . . 10

2.5.2 Different properties of surface plasmon . . . . 12

2.6 Principle of 2D optical waveguides . . . . 18

2.6.1 Film SPP modes . . . . 18

2.6.2 Gap SPP modes . . . . 21

3 Comparison of different plasmonic waveguides 23 3.1 Definition of confinement, propagation length and figure of merit . . . . 24

3.1.1 Effective modal area . . . . 24

3.1.2 Propagation length . . . . 25

3.1.3 Figure of merit . . . . 25

3.2 Slot waveguides . . . . 26

3.2.1 Simplified geometries . . . . 26

3.2.2 Characteristic of symmetric slot waveguide . . . . 27

3.3 Metal strip waveguides . . . . 33

3.4 Metal cylinder waveguides (CyPPs) . . . . 35

3.5 Dielectric-loaded SPP waveguides . . . . 36

3.6 Hybrid plasmonic waveguides . . . . 38

3.6.1 Conductor gap dielectric . . . . 38

3.6.2 Hybrid plasmonic waveguide with a metal cap . . . . 43

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4 Waveguide design and analysis 47

4.1 Geometry and dispersion characteristics . . . . 47

4.2 Confinement and propagation . . . . 49

4.2.1 Effect of gap width . . . . 51

4.2.2 Effect of nanowire diameter . . . . 51

4.2.3 Effect of strip width . . . . 52

4.3 Power distribution profile . . . . 52

4.4 Analysis of hybrid mode . . . . 54

4.5 Coupling between two parallel waveguides . . . . 54

4.6 Figure of merit . . . . 57

5 Optical trapping 59 5.1 Force in hybrid Si nanowire . . . . 59

5.1.1 Force with propagation loss . . . . 60

5.1.2 Nanoparticle trapping . . . . 61

5.2 Force in hybrid Ge nanowire . . . . 63

5.2.1 Force with propagation loss . . . . 64

5.2.2 Nanoparticle trapping . . . . 64

5.3 Force in plasmonic metal nanowire . . . . 64

5.4 Force in hybrid Ge strip novel waveguide . . . . 64

6 Conclusion 67

A Matlab script to calculate mode area A 3 69

Chapter 1

Introduction

It is now routine to produce ultrafast transistors with sizes on the order of 50 nm. Scaling blessed them in terms of power consumption and speed of operation but also brought delay in terms of interconnections [1]. This delay imposes a fundamental limit in the data transfer capacity that an electronic chip can support, which in turn drives us into the research for increasing data transfer capacity along with a high integration density. Although optical interconnects are capable of providing very high bandwidth in terms of data transmission, their downscaling in size is impeded by the diffraction limit [2]. An ideal solution to this problem would be to have a circuit which supports both optical signals and electric currents.

The fascinating field of nanophotonics addresses the challenge of manipulating light over di- mensions on the order of or smaller than the wavelength. Surface plasmon (SP) based photonic circuits are the most promising candidates for achieving high density photonic integration.

They offer the potential to have the capacity of photonics and the small dimension of elec- tronics. Surface plasmon (SP) waves are collective oscillation of electrons, moving back and forth near the surface of metal/dielectric [3]. When light comes to interact with the plasma wave, coupling between them occurs and surface plasmon polaritons (SPP) are formed. The decay lengths (skin depths) of SPP waves in the direction transverse to the interface, are very small which make SPP a strong candidate to offer sub-wavelength mode confinement and possibility to guide light in metallic nano-structures. Plasmonic circuits will bridge the gap between photonics and electronics. They potentially serve as building blocks for various optical components such as waveguides, couplers, switches etc.

So far there have been a lot of theoretical proposals and experimental demonstrations of SPP waveguides. V-shaped grooves in metals [4] and wedge structures [5] are proposed and experimentally demonstrated to achieve localized field enhancement. High degree of confinement of the plasmonic circuits comes with the cost of high loss inherited due to the use of metal. An array of nanoparticle resonators can be used to reduce ohmic losses [6]. Hybrid waveguides are proposed to provide long propagation along with sub-wavelength confinement [7]. There is also experimental demonstration of deep subwavelength scale nano laser using

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capacitor-like energy storage of the hybrid mode [8].

Although there are lot of promising structures available, there is a lack of cross-comparison among these waveguides in the literature. Also the trade-off between the propagation and confinement is yet to be solved/improved. It is one of the objectives of this thesis to consis- tently investigate the modal properties for different structures and compare them in terms of merit.

In this thesis work a variety of plasmonic structures are analysed and their modal charac- teristics have been cross-compared. Different definitions of confinement are studied and a graphical way to measure the figure of merit has been demonstrated. A new waveguide based on hybrid mode coupling has been proposed. The characteristics of the new design is thor- oughly investigated in order to compare with the state-of-the-art waveguide sturctures in the literature.

Full vectorial eigenmode solver of COMSOL has been used to calculate the effective indices of the modes. At the extremities of the calculation region perfectly matched layers (PML) along with scattering boundary condition is used to mimic open boundary. Convergence analyses are also been conducted to verify the validity of the results.

An overview of the content of the chapters is given below,

Chapter 2 : Principles of plasmonic waveguides are studied and several important features of SPP waves are reviewed.

Chapter 3 : Various plasmonic structures are theoretically investigated and several charac- teristics parameters are calculated. Comparing structures using graphical figure of merit is introduced.

Chapter 4 : A novel design is proposed and the characteristics of the guided mode are anal- ysed.

Chapter 5 : Theory of optical force is introduced. Optical trapping forces for several plas- monic structures are calculated including the proposed structure with air background.

Chapter 6 : Summary of the work is presented.

Chapter 2

Principle of surface plasmon waveguiding

2.1 Introduction

In order to realize plasmonic or the subwavelength optical waveguides, we first need to un- derstand surface plasmon polaritons (SPPs). Polaritons are considered as quasi-particles resulting from strong exchange of energy between electromagnetic wave and excitation in a material e.g. photon-electron coupling. When electromagnetic fields are coupled to the oscil- lation of electron plasma of a conductor in a dielectric-conductor interface, electromagnetic surface waves are excited and propagate along the interface. These surface waves are evanes- cently confined in the perpendicular direction and are known as surface plasmon polariton waves [9]. SPP waves in metallic waveguides and metal nanostructures open the possibility to confine and guide optical waves on the nanometer scale [10].

Metal waveguides are also known as negative dielectric waveguides because the real part of their permittivity  is negative at optical frequencies (from visible to infra red). In recent years, we have seen metamaterials having both negative permittivity and permeability [11], which we will define as negative index (NI) materials. Optical waves can propagate through dielectric (having, Re[] > 0 and Re[µ] > 0 ) and NI materials. However propagation of optical wave is prohibited in materials having either negative  or negative µ (ND and NP).

Thats why we need a new approach to guide waves in ND based waveguides. We can excite self-sustaining SPP mode which will propagate along the boundary between ND and dielec- tric. The categorization of different materials in terms of the signs of  and µ is shown in Figure 2.1. We will consider ’ND’ and ’metal’ both having same meaning and will use them interchangeably.

Starting from the wave equation, this chapter introduces the concept of propagating and evanescent waves, low dimensional optical waves, diffraction limit of conventional optics and

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Figure 2.1: Position of various materials in the  − µ diagram in terms of both signs of  and µ.

Propagation is prohibited in Negative Dielectric (ND) and Negative Permeability (NP) materials (shaded areas).

different properties of SPPs. We will concentrate on the underlying physical phenomena involved in surface Plasmon waveguides and analyze various 2D and 1D waveguide configu- rations.

2.2 The electromagnetic wave equation

As SPPs are electromagnetic (EM) waves, in order to study propagating SPP modes we need to apply Maxwell’s equations to the flat interface between a conductor and a dielectric. We can start from a general form applicable for the guiding of any EM waves, the wave equation, which can be used to determine spatial field profile and dispersion of propagating waves eventually.

Maxwell’s equations for macroscopic electromagnetism [12]:

∇ × E = − ∂B

∂t ,

∇ × H = J ext + ∂D

∂t ,

∇ · D = ρ ext ,

∇ · B = 0,

(2.1)

in which the charge density ρ _ext and the current density J _ext are the sources for the electric

field E and the magnetic field H. After combining the curl equations, EM waves propagating

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in a nonmagnetic, isotropic, homogeneous medium, in the absence of external charge and current densities, we can get,

∇ × ∇ × E = −µ 0

∂ ² D

∂t ² (2.2)

If the dielectric profile is invariant over distances in the order of single optical wavelength, we can simplifies (2.2) to,

∇ ² E −  c ²

∂ ² E

∂t ² = 0 (2.3)

After assuming harmonic time dependence E(r, t) = E(r)e ^−iωt of the electric field,

∇ ² E + k ² ₀ E = 0 (2.4)

This is known by Helmholtz Equation where, k ₀ = ^ω _c is the propagating wave vector in vacuum. Now, let’s define the propagation geometry under the simplicity of one dimensional problem as shown in Figure 2.2, where permittivity  depends only on one spatial coordinate.

If the propagation direction is x in the Cartesian coordinate system, no variation along y axis (in-plane) and z = 0 defines the interface, we can write  = (z) and the propagating wave can be described as E(x, y, z) = E(z)e ^iβx . The component of the wave vector along the propagation direction, k _x is known as the propagation constant β of the traveling wave. After inserting this expression into (2.4),

∂ ² E

∂z ² + (k ² ₀  − β ² )E = 0 (2.5)

A similar expression for the magnetic field H, can also be written [9]. Now we need to know the explicit equations for different field components of E and H in order to determine the spatial field profiles and dispersion relations. For TM modes where electric fields has non zero components on the plane parallel to propagation (p polarized), only the E _x , E _z and H _y components exist and the wave equation for TM mode is,

Figure 2.2: Planar waveguide geometry and propagation of waves in a cartesian coordinate system.

∂ ² H y

∂z ² + (k ² ₀  − β ² )H y = 0 (2.6a) with electric field components,

E x = −i 1 ω ₀ 

∂H y

∂z (2.6b)

E _z = − β

ω 0  H _y (2.6c)

For TE modes where electric field has the only nonzero components along perpendicular to the propagation plane (s polarized) we have H x , H z and E y being nonzero and the wave equation for TE mode is,

∂ ² E y

∂z ² + (k ² ₀  − β ² )E y = 0 (2.7a) with magnetic field components,

H x = i 1 ωµ 0

∂E y

∂z (2.7b)

H z = β ωµ 0

E y (2.7c)

2.3 Propagating and evanescent waves

The SPP modes supported by metal nanostructures comprise exponentially decaying evanes- cent fields and confined in a plane perpendicular to propagation direction. Therefore, it is useful to review the properties of propagating and evanescent waves at the boundary between different media [2, 13]. If we substitute plane-wave trial solution of electric field in the form E(r, t) = E 0 exp [i(k.r − ωt)] in Eq.(2.3) we can have the dispersion relation for the wave vector k(ω) as,

|k(ω)| ² = k ² = k _x ² + k _y ² + k ² _z = (ω) ω ²

c ² = k ² ₀ (2.8)

Considering 2D geometry and xz as propagating plane, no variation along y axis (k y = 0), field equation for a plane wave propagating in the half-space z ≥ 0 becomes,

E(r, t) = E 0 exp ( −iωt) exp [i(k ^x x + k z z)] (2.9)

with k x is in ( −∞, ∞), and k z = pk ₀ ² − k ² x

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Figure 2.3: Example of the occurrence of evanescent waves, by (a) total internal reflection in dielectric when the angle of incidence is more than some critical angle, (b) reflection for any angle of incidence when the material is metal.

Now it is intuitive from the above relation that all waves with k x ≤ k 0 are propagating or homogenous wave and waves having k _x > k ₀ are defined as evanescent waves [14].

If k x ≤ k 0 , both the wave vector projections (k x and k z ) are real and propagation direction of propagating waves is determined by this. They have wavelength λ = 2π/k = λ 0 /n, where λ ₀ = wavelength in vacuum and n ² = . For the case of evanescent waves propagating along the x-axis, the spatial period of oscillation is λ x = _|k ^2π

x

| . As they are nearfield standing waves and decays exponentially from the boundary along the z-axis, the decay length is given by

| k z | ⁻¹ = (k _x ² −k 0 ² ) ^−0.5 The spatial periodicity of this kind of waves can go infinitely small (i.e.

spatial frequency extremely high) [14]. So we can realize their use in nanophotonic circuits in general and interconnections in chip, in particular.

Evanescent waves can be formed in dielectric by total internal reflection of light as shown in Figure 2.3. They exist because the tangential component of electric and magnetic field cant be discontinuous (i.e. phase matching of the total electric fields on both sides of the interface) at the boundary, as would be the case if they were not present. When the angle of incidence θ > θ cr (determined from √ d sin θ cr = 1) of a wave propagating in the half space z < 0 in a dielectric with  _d > 1 and strikes the interface z = 0 between dielectric and air with  _air = 1, an evanescent wave will be excited in the half-space z ≥ 0, because k x >  _air k ₀ = k ₀ and to fulfill the boundary condition,

k _x = √  _d k ₀ sin θ > k ₀ ⇒ E = E 0 exp (i(k _x x − ωt)) exp



−zk 0

q

 _d sin ² θ − 1



(2.10)

Evanescent waves can be excited at any angle of light incidence if one of the medium is metal.

For the case of a lossless metal with  =  m < 0, the wave number k is pure imaginary and there are no propagating waves. Considering half-space z > 0 occupied by the metal, we can find that any propagating wave from air to metal will form an evanescent wave in metal.

As for typical metals, |  ^m | 1 [15], the field penetration (decay length) in metal |k ^z | ⁻¹ ≈

λ ₀ /(2π | m | ^0.5 ), is actually very small, typically in the scale of nanometers. We will show in

section 2.5 that contrary to propagating wave, evanescent waves can only be excited with TM

polarization in order to satisfy Coulomb’s law.

2.4 Diffraction limit and low dimensional optical waves

Nano-optics is different from conventional optics in the sense that, the exponentially decaying spatial field plays an essential role in Nano-optics, on the other hand, in conventional optics it is the extended mode which plays the dominating role. The dimension of an optical wave is defined as the number of real components in wave vector k. In Cartesian coordinate system if an optical wave have three real components in k then it is called a 3-dimensional (3D) optical wave. Optical waves having one or more imaginary components of k are low dimensional waves [16]. If the optical wave doesn’t have any real component of k then it is defined as zero-dimensional (0D) optical wave. In this section we will model optical waves with exponentially decaying field based on the concept of low dimensional optical waves.

2.4.1 Diffraction limit in 3D waves

For a dielectric optical waveguide it is the diffraction limit that will define the level of con- finement that can be achieved during wave guiding. Considering a plane wave with angular frequency ω in a medium with refractive index n, the dispersion relation of light will be

| k | ² = n ² ω ²

c ² (2.11)

In Cartesian coordinate it becomes,

k ² _x + k _y ² + k _z ² = (nk ₀ ) ² = µ( ω

c ) ² (2.12)

Figure 2.4: Plot of equation (2.12) : wavenumber (k) surface [17] for different dimensional waves in

different materials, (a) 3D optical wave in dielectric, (b) 2D optical wave in dielectric,

(c) 2D optical wave in metal (ND), Ref. [18].

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Since in 3D optical waves all the components of wave vector are real, the values of each components k _j are in the range −k ≤ k j ≤ k where j = x, y, z. So the spatial frequency

∆k = 2k = 2nk ₀ where k ₀ = ^2π _λ

0

is the wavenumber in vacuum. From the uncertainty relation of fourier transform, ∆r∆k ≥ π where ∆r = range of real space. Because of this constraint,

∆r has a minimum,

∆r ≥ π

∆k = λ ₀

4n (2.13)

So the minimum size of a confined beam in all direction for a 3D optical wave,

∆r _min3D = λ 0

4n (2.14)

This is true for all waveguides irrespective of their structures, shapes and materials as long as the optical wave is 3D (i.e. all components of wave vector k are real). Even for photonic crystal or higher refractive index waveguide, the minimum size of the beam is limited by diffraction to the order of vacuum wavelength (λ 0 ).

2.4.2 Diffraction limit in 2D waves

From the study of propagating and evanescent waves in the previous section we saw that wave vector along the perpendicular direction was imaginary for the case of evanescent waves.

So, 2D optical waves can be generated with the help of total internal reflection in dielec- tric/dielectric interface and k components (k x , k y , ik z ) follows the relation,

k ² _x + k _y ² − k z ² = µ( ω

c ) ² , (2.15)

where, (> 0) is the permittivity of the medium where the evanescent wave has been generated.

If we plot this equation we get the surface of one-sheeted circular hyperboloid (1SCH) in 2D k-space as shown in Figure 2.5(a). In the figure we considered two wavenumber surfaces on each side of the interface and they must coincide due to the boundary condition.

For the case of evanescent wave at dielectric/metal interface using the same equation we can

plot the surface again using  < 0 which is shown in Figure 2.5(b). Here, this is the surface of

two-sheeted circular hyperboloid (2SCH). For Figure 2.5(a) and (b) there is a closed sphere

since we still need 3D optical wave at one side of the interface, ∆k is limited by the radius

of the sphere. For 2.5(c) however, surface plasmon polariton do not need to be accompanied

with 3D wave. We have 2D optical wave at both interface → wavenumber surfaces are 2SCH

at metal side and 1SCH at dielectric side. Since, surface topology is open for both 1SCH and

2SCH ∆k can be increased indefinitely and ∆r → 0 when ∆k → ∞. However there is still a

limit even for the case of SPP, which we will discuss more in the next section.

Figure 2.5: Wavenumber (k) surface along with applied boundary condition for different materials interfaces, (a) Total internal reflection at dielectric/dielectric, (b)Reflection at dielec- tric/metal, (c) Surface plasmon polariton at dielectric/metal interface, [18]

2.5 Surface plasmon polariton

2.5.1 Dispersion relation of SPP at a single interface

Let us consider the most simple geometry sustaining SPP on a plane dielectric/metal interface as shown in Figure 2.6. The non-absorbing half space (z > 0) consists of dielectric with positive real dielectric constant  _d and the adjacent ND half space (z < 0) is described via a dielectric function  _m (ω).

Figure 2.6: Domain and coordinate considered for SPP propagation at a single interface.

We will first consider TM field (H y , E x and E z ) solutions. If we seek for bounded wave confined to the interface, for domain z > 0 requires β ² − k ² 0  _d > 0 i.e. evanescent decaying away from the interface in the perpendicular z-direction. So the solution is [9],

H _y (z) = A exp(iβx) exp( −K 1 z) (2.16a) E x (z) = iAK 1

ω ₀  _d exp(iβx) exp( −K ¹ z) (2.16b)

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E z (z) = − Aβ

ω 0  _d exp(iβx) exp( −K 1 z) (2.16c) Here, A is a constant and,

K 1 = q

β ² − k ² ₀  d (2.17)

for z < 0 similarly,

H y (z) = B exp(iβx) exp(K 2 z) (2.18a) E x (z) = − iBK 2

ω ₀  _m exp(iβx) exp(K 2 z) (2.18b)

E z (z) = − iBβ ω 0  m

exp(iβx) exp(K 2 z) (2.18c)

Here B is another constant and,

K ₂ = q

β ² − k ² 0  _m (2.19)

K _i ≡ k z,i (i = 1, 2) is the component of the wave vector perpendicular to the interface in the two different media. Note that, when the operating frequency is below the metal plasmon frequency,  m is negative, and argument of the square root will always fulfill the condition β ² −k ² 0  _m > 0. Now we have to apply the boundary condition in order to form a single surface wave. From the continuity of displacement and electric field,

D ¹ _⊥ = D ² _⊥ (2.20a)

E _k ¹ = E _k ² (2.20b)

at the interface z = 0 between the D and ND media. From the first boundary condi- tion(continuity of D _z ⇒  d E _z =  _m E _z ), we can obtain,

A = B (2.21)

and from second boundary condition (continuity of E x ), AK ₁

 d

+ BK ₂

 m

= 0 (2.22)

which can be written with the help of the other, K ₁

 d

+ K ₂

 m

= 0 (2.23)

Using this along with the expression for K 1 and K 2 we arrive at the central result of this section,

β = ω ² c ²

r  d  m

 _d +  _m (2.24)

This is the anticipated dispersion relation of SPP as it contains both ω and β. This expression is valid for both complex and real valued  _m (ω) i.e. metal with and without attenuation. Eq.

(2.24) implies that the effective permittivity that SPP ’feels’ is _ ^

^

d

+

, while the condition of propagation is,

 _m < − d < 0 = ⇒ | m | >  d (2.25) Before going deep into the properties of the dispersion relation, let’s analyse the possibility of TE modes. Using (2.9), we can find the field components in the similar way we found for TM. for z > 0

E _y (z) = A exp(iβx) exp( −K 1 z) (2.26a) H _x (z) = − iAK ₁

ωµ 0

exp(iβx) exp( −K 1 z) (2.26b)

H _z (z) = Aβ ωµ 0

exp(iβx) exp( −K 1 z) (2.26c)

for z < 0

E y (z) = B exp(iβx) exp(K 2 z) (2.27a) H x (z) = iBK 2

ωµ 0

exp(iβx) exp(K 2 z) (2.27b)

H z (z) = Bβ ωµ 0

exp(iβx) exp(K 2 z) (2.27c)

Again from the boundary condition (continuity of E _y and H _x ) at the interface, we may obtain,

A(K 1 + K 2 ) = 0 (2.28)

From the condition of the propagation of the surface modes Re[K 1 ] > 0 and Re[K 2 ] > 0. So in order for the confinement to the surface, A = 0, also B = A = 0. Thus, TE polarization cannot sustain any surface modes. Existance of surface plasmon polaritons only possible with TM polarization [9].

2.5.2 Different properties of surface plasmon

Now lets take a closer look at the dispersion relation. If we assume the upper half domain is air ( d = 1) and the lower half domain is metal with negligible damping (ωτ  1, i.e. lossless) described by the real Drude dielectric function,  _m = 1 − ω p ² /ω ² . We can write the dispersion equation in a more explicit way,

β ² c ² ω _p ² =

ω

ω

 ω

ω

− 1  2 ^ω _ω

p

− 1 (2.29)

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Figure 2.7: Dispersion relation of a typical SPP confined by a Drude metal with negligible damping frequency and air (blue curve) and silica (red curve) interface. The dotted line is for imaginary part of β. There are three regions in this graph - top left is (ω > ω p ) radiation region, curve lying to the right of the respective light line are the bound mode region and in between them where β is purely imaginary, is the frequency gap region.

Figure 2.7 shows the plot of (2.29) for lossless Drude metal with both an air ( _d = 1) and a fused silica ( d = 2.25) interface. In this graph the frequency ω and wave vector β are normalized. The real and imaginary part of β is shown with continues and broken curves respectively. In the frequency range ω < ω _p , we have  _m < 0 which fulfills the assumption β ² −k ² 0  m > 0. So β is real in the frequency region with ω/ω p between 0 and 0.71 for air-metal and between 0 and 0.54 for silica-metal interface which ensures the wave is propagating. As the SPP excitations always correspond to the curves lying below of respective light lines, special phase-matching techniques are necessary for their excitation.

For small wave vectors corresponding to low frequencies, the SPP propagation constant β coincides with the light line quite well. Since, in this regime the waves penetrate over many wavelengths into the dielectric they are also known as Sommerfeld-Zenneck waves [19] and are more photon like.

The upper limit of frequency for propagating mode is defined as the surface plasmon frequency [2],

ω _sp = ω _p

√ 1 +  _d (2.30)

Here,  _d is 1 for air and 2.25 for silica. So, ω _sp,1 = 0.7071ω _p for air-metal and ω _sp,2 = 0.5547ω _p for silica-metal interface. At frequency close to ω sp dispersion curve becomes more flat and approaches β → ∞, if  d +  m (ω) = 0.

In Figure 2.7, a lossless metal with dielectric function given by Drude model has been assumed

with Im[ m ] = 0. However in real metals  m (ω) is complex and so is the propagation constant

Figure 2.8: Dispertion relation of SPP confined by a Silver/Air (blue curve) and Silver/Silica inter- face (red curve) after considering damping. At surface plasmon frequency propagating wave vector approaches a finite value in this case.

β, because of free electron and inter-band damping. Using the data for  m (ω) of silver (Ag) obtained by Johnson and Christy [15], we plot the dispersion relation of SPP at silver/silica and silver/air interface in Figure 2.8. After we consider damping, the wave vector approach a maximum, finite value when the frequency is close the surface plasmon frequency ω sp of the system, instead of having something close to infinity for the case of undamped SPP. Also the energy of traveling SPPs are attenuated by e ⁻¹ after some distance defined as the propagation distance, L p = _2Im[β] ¹ . This is because of the ohmic losses of the electron oscillation associated with damping which limits both the propagation length and extent of decrease of λ _sp (= 2π/β).

Also in contrast to the case of ideal metal, the region between ω _sp and ω _p is not forbidden any more, rather a quasibound leaky part of Re[β] exist in this regime [9].

A limited SPP propagation length is the main difficulty behind developing surface plasmon based waveguide components for real life applications. For an example, we can estimate the normalized propagation length at silver-air interface using permittivity of silver,  m =

−129 + 3.3i at the telecom wavelength of 1.55 µm obtained from Johnson and Christy [15].

After putting the values in (2.24) we get L p /λ sp ≈ 796 which essentially represents the upper limit of merit factor for the SPP-based resonators and interferometers.

Group Velocity

Using Eq.2.29 we can obtain the expression for group velocity as,

v _g = dω dβ =

 2 ^ω _ω

p

− 1 

2



ω

ω

− 1 

2

2 

ω ω

 4

− 2 ^ω _ω

+ 1

c (2.31)

15

Figure 2.9: Group velocity as a function of normalized frequency in the unit of c. Blue curve for air and red curve for silica as dielectric.

The plot of the group velocity in Figure 2.9 shows that as the frequency approaches ω sp it decreases from c to zero. When v _g → 0, the mode acquires electrostatic character, thus it forms as surface plasmon.

Field property and enhancement

From the boundary condition, the tangential component of electric field (E x ) is same both in dielectric and metal(E _x ⁰ ). Since the field components need to satisfy Coulomb’s law ∆.E = 0 both in upper and lower half spaces, we can explicitly relate the amplitudes of normal (perpendicular to the interface) and tangential field components in the dielectric and metal with the help of dispersion relation [Eq.(2.24)],

E z,d = i r − m

 d

E _x ⁰ and E z,m = −i r − d

 m

E _x ⁰ (2.32)

Again, typically | m |   d , resulting the domination of E z (normal component) in the dielec- tric and and E x (tangential component) in the metal [14]. This reflects the hybrid nature of SPPs combining the features of propagating EM wave (with transverse electric field, E _z ) in dielectrics and free electron oscillations (with longitudinal electric field, E x ) in metals. Also, SPP damping due to ohmic losses in metal, is determined from the longitudinal electric field components of the SPP in the metal.

An SPP is able to provide a much higher electric-to-magnetic field ratio than the ratio in free

space plane wave propagation. This property of SPP is known as field enhancement which

can lead to many nonlinearity related applications such as surface enhanced Raman scaterring

(SERs).

According to Eqs. 2.16c and 2.16a, the ratio between the field components is, E _z

H _y = β

ω ₀  _d = β

k ₀  _d Z ₀ =  _m

 _d +  _m Z ₀ (using Eq. 2.24) (2.33) If we assume the upper half-space is free space ( _d = 1) and lower domain is Drude metal with  m = 1 − ω ² p /ω ² , then the field ratio as a function of frequency ω,

E z

H _y =

ω

ω

− 1 2 ^ω _ω

p

− 1 Z 0 (2.34)

Relating to nonlinearity which depends on E ² , we can define the enhancement factor as,

f =





ω

ω

− 1 2 ^ω _ω

p

− 1





2

(2.35)

Figure 2.10: Electric field intensity enhancement at a metal surface compared to in free space.

It is quite evident from the graph in Figure 2.10 that, it is desirable to operate at frequency close to ω sp to order to achieve significant enhancement at the surface. The other component tangential field E _x , also experiences similar enhancement effect.

Diffraction Limit of SPP

When the condition of SPP propagation is met ( | ^m | >  d ), one can see that β = k spp is larger than nk ₀ ,

k spp = nk 0

1 q

1 − | ^|

^| |

> nk 0 , (2.36)

17

(a) (b)

(c)

Figure 2.11: (a,b) Wave number (k SP P ) surface with respect to permittivity of metal in complex plane (a) Air-Metal, (b) Silica-Metal (c) Width of SPP (solid red curve) with respect to the relative permittivity of lossless metal for  d = 1. Decay length in dielectric (D) (dotted blue curve) and metal (ND)(dash-dotted green curve) are also shown.

where n is the refractive index of the dielectric (D) medium defined as n = √ _d . As EM field of SPP is confined as a 2D optical wave at the interface z = 0, we can find the width of SPP by adding decay length (for the 1/e amplitude decrease) in both D and ND media.

From section (2.3) we know the decay length for evanescent wave is given by | k z(d,m) | ⁻¹ = (k _x ² −  d,m k ² ₀ ) ^−0.5 , where k _z(d,m) is the wave vector in the perpendicular direction of D and ND medium respectively.

δ SP P = δ d + δ m = λ 0

2π ( p| m | −  d

 _d + p| m | −  d

| ^m | ), (2.37)

so that  _d δ _d = | m |δ m . Since,  _m   d , the field penetration in the metal is typically much shorter than that in the dielectric.

To estimate the typical value of this width, we may use silver (Ag) as typical ND ( m =

−129 + 3.3i) and Air or Silica as typical D. As a numerical example for silica( d = 2.25),

δ d = 12.183 µm, δ m = 0.2125 µm, k SP P = 1.5133 k 0 and for air ( d = 1) δ d = 27.55 µm, δ m =

0.2135 µm, k SP P = 1.0039 k 0 at λ = 1550 nm and lossless values of ND are considered. If we consider damping in metal as well, these values become, for silica, δ _d = 2.7776 µm, δ _m = 0.0215 µm, and for air δ _d = 2.7912 µm, δ _m = 0.0216 µm. In Figure 2.11(a) and 2.11(b) we can see that k SP P diverges to infinity as | m | →  d . So, its clearly evident that k SP P can be increased by changing material parameter of D or ND. From Figure 2.11(c), the width of SPP (δ SP P ) can be decreased rapidly when  m ≈ − d . The minimum size of a beam that can be synthesized can be written as,

∆r min2D (SP P ) = λ _{SP P} 4 = λ ₀

4n

r  d +  _m

 m

(2.38) where, n = √ _d and because  _m < − d < 0 we can reach the conclusion that ∆r _min2D for SPP is smaller than ∆r _min3D ,

∆r min2D (SP P ) = λ ₀ 4n

s

1 − | d |

| m | < λ ₀

4n (2.39)

In this way we can overcome the limit of diffraction of 3D optical waves by using SPP.

Nevertheless the minimum size is still limited to the order of subwavelength and depends on material constants. It would be possible to make ∆r min2D much smaller if we could somehow increase k without changing material parameters. The coupled system of SPP is promising to do that.

2.6 Principle of 2D optical waveguides

Let us move on from single interface to multiple interface systems. When we place two metal/dielectric interfaces close together, the SPPs associated with individual interfaces start to interact with each other, it is known as the coupled system of SPP. Considering the SPP modes associated with two metal/dielectric interfaces, we can have either a thin metal layer sandwiched between two layers of dielectrics or a thin dielectric layer surrounded by metals.

These structure are shown as metal film and metal gap in Figure - 2.12(a) and 2.12(b) respectively.

2.6.1 Film SPP modes

Let us consider the SPP modes in the symmetric dielectric-metal-dielectric configuration as

shown in Figure-2.12(a). Here, a thin metal film with thickness h is embedded in the dielectric

which extends indefinitely on both side of the metal. Electromagnetic field is assumed in the

form e ^i(ωt−βx) while the propagation is along x direction. If the thickness is narrow enough

the two identical SPP mode at both interfaces will start to overlap with each other. The

propagation constant of individual symmetric and antisymmetric mode is different for small

thickness of the metal film [20,21]. Since transverse and longitudinal components complement

each other, symmetric component of one would be antisymmetric of the other configuration.

19

(a) (b)

Figure 2.12: Coordinate and geometry of 2D optical waveguides with double metal/dielectric inter- face. The propagation is along x direction. (a) Metal film with dielectric on both side, (b) Metal gap with dielectric between two metal slab.

Conventionally E _z -symmetric configuration is known as the symmetric SPP mode (exhibit odd symmetry of E x ). In this case E x changes its sign at the mid-plane of the metal film and experiences considerably less attenuation than the anti symmetric SPP mode. This symmetric branch in the metal film, which has long propagation length even in lossy metals, is called the long-range SPP (LR-SPP) and the anti-symmetric one (symmetric E x ) is known as - the short range SPP (SR-SPP) [21, 22]. After applying appropriate boundary condition, we obtain the characteristic equations of LR-SPP for TM modes of 2D optical waves in metal film,

tanh(k _z ^(m) h/2) = −  _m k z ^(d)

 _d k ^(m) z

where, k _z ^(m,d) = q

k _LR−SP ² −  m,d k ² ₀ (2.40)

Although LR-SPP propagation constant cannot be expressed in an explicit form like SPPs, for thin film with h → 0, we can approximate tanh x ≈ x and present k LR−SP in the following way [14],

k LR−SP ≈ k 0 p d + (hk 0  d /2) ² · [1 − ( d / m )] ² (2.41) For very thin film, LR-SPP propagation constant approaches the light line in the dielectric.

The main component E x crosses zero at the middle of the metal and spreads out to dielectric as shown in Figure 2.13(a). This reduces propagation loss due to damping and make LR-SPP suitable for developing components for integrated photonics.

For SR-SPP, the main component in metal, E x is nearly constant and component in dielectric,

(a) (b)

(c) (d)

Figure 2.13: Electric field distribution of gold film surrounded by air, film thickness h = 50 nm and operating at λ 0 = 775nm. Perfect magnetic conductor (PMC) has been applied to the side boundaries and scattering boundary condition is applied on top and bottom boundaries, while the propagation is at x direction. (a) Long Range E x (changes sign at the mid plane),(b) Long Range E z (symmetric),(c) Short Range E x (nearly constant in the metal),(d) Short Range E z (antisymmetric).

E z is anti-symmetric as shown in Figure 2.13(c) and 2.13(d) respectively. Following the same procedure the dispersion relation of SR-SPP is [14],

tanh(k ^(m) _z h/2) = −  _d k ^(m) z

 _m k ^(d) z

where, k _z ^(m,d) = q

k ² _SR−SP −  m,d k ² ₀ (2.42)

and simplification after assuming very thin films,

k _SR−SP ≈ k 0 p d + [2 _d /(hk ₀  _m )] ² (2.43)

When the film thickness h → 0, k SR−SP increases indefinitely : k SR−SP ≈ −(2 d )/(h m ) →

∞, resulting both the SR-SPP wavelength and propagation length approaching zero. Due to

the high reflectivity at the film termination exhibiting by SR-SPP they are useful for designing

various plasmonic resonator structures [23].

21

Figure 2.14: Effective index and propagation length of long and short range SPP modes for a gold film ( m = −23.6 + 1.69i) surrounded by air at the excitation wavelength of 775nm (adapted using Ref. [24] )

2.6.2 Gap SPP modes

Now lets consider the SPP modes in the metal-dielectric-metal configuration as shown in Figure-2.12(b). The propagation mode in the metal gap split into two branches as the gap decreases. The mode exhibiting odd symmetry of E x (longitudinal component) survives for all the values of h and is known as Gap SPP (G-SPP). Using boundary condition and symmetry of field components, we can obtain the dispersion relation of G-SPP [14],

tanh(k _z ^(d) h/2) = −  _d k ^(m) z

 m k z ^(d)

where, k _z ^(m,d) = q

k _G−SP ² −  m,d k ₀ ² (2.44)

Here, k G−SP is the propagation constant for the fundamental G-SPP mode and E z has the same sign across the gap (Figure 2.15(b)) while E _x (Figure 2.15(a)) has opposite sign.

Simplification after assuming sufficiently small gap, h → 0,

k _G−SP ≈ k 0

r

 _d + 0.5(k _G−SP ⁰ /k ₀ ) ² + q

(k ⁰ _G−SP /k ₀ ) ² ( _d −  m + 0.25(k ⁰ _G−SP /k ₀ ) ² ) (2.45)

where in the limit of very narrow gaps, Gap-SPP propagation constant, k _G−SP ⁰ = − _h ^2

. The fractional electric field energy concentrated in the gap region increases (damping decreases) with the decrease of gap width when the gap is larger. After reaching the maximum it starts to decrease with the field squeezing into the metal (damping increases). So the propagation length can increase when h decrease from large values which corresponds to uncoupled SPPs.

This remarkable fact of G-SPP makes it possible to have modes with better confinement with

longer propagation length.

(a) (b)

Figure 2.15: Electric field distribution of two gold (Au) slab with an air gap of thickness h = 50nm,

(a) Gap-SPP E x (odd symmetry/opposite sign across the gap), (b) Gap-SPP E z (have

same sign across the gap).

Chapter 3

Comparison of different plasmonic waveguides

The need of optical components capable of guiding and manipulating light in the sub- wavelength scale has motivated research to plasmonics. In the past few years we have seen a surge in sub-micron scale devices. Although their physical dimension is smaller than the wavelength in vacuum, their confinement capability is rather limited by diffraction. Even the promising structures like photonic crystals [25] and all-dielectric coupled silicon waveg- uides [26] have fundamental limit in terms of diffraction. In the previous chapter we have shown, it is possible to break this limit with the help of low dimensional evanescent waves. In the visible regime the typical way to achieve sub-wavelength confinement involves the use of surface plasmon polaritons (SPPs). There are numerous waveguide architectures available in the literature which utilizes the concept of SPP by storing part of light’s energy as electron plasma oscillations at the interface between a metal (ND) and dielectric (D). The decay length of the bound modes are typically much smaller than the wavelength. The SPP modes are capable of squeezing light into sizes smaller than the diffraction limit but they have limited propagation length because of intrinsic losses imposed by the use of metals.

From the literature we can find various plasmonic structures having extreme confinement such as metallic nanowires [27], metallic nanoparticles [28], V-shaped grooves [4] and wedges [5].

These geometries are limited by the fact that they support highly confined mode only when operating near the surface plasmon frequency (ω _sp ). Also the propagation length is in the range of few µm. Although there haven a lot of theoretical proposals and experimental demonstration have been made, there is a lack of cross-comparison among these waveguides in the literature. This is one of the goal of this thesis work to find a consistent way to measure the modal properties. We will discuss, analyse and compare the performance of several plasmonic waveguide structures namely, metal slot waveguide [29], metal stripe [30], dielectric loaded SPP waveguide [31], metal cylinder (CyPP) [32], hybrid metal cap [33] and hybrid dielectric cylinder structure [7]. We have thoroughly investigated the properties of the modes supported by these structures and examine their merits in terms of confinement and

23

propagation. We used a commercial package, the RF module of COMSOL Multiphysics for simulations. The full vectorial finite element eigenmode solver is used to find modes of the waveguide. At the extremities of the calculation region perfectly matched layers (PML) along with scattering boundary condition is used to mimic open boundary. Convergence analyses are also been conducted in terms of calculation region so that the effective indices do not vary by more than 1%. Initial guess of index is always chosen to be higher than the highest index.

3.1 Definition of confinement, propagation length and figure of merit

3.1.1 Effective modal area

The available measures of modal area are inherited from conventional waveguide theory. Since the plasmonic waveguide can have sharp features which lead to rapid sub-wavelength level variations in the shape of the mode, these measures are somewhat inconsistent if we apply them for plasmonic mode. We need to find a definition which will consider the true extent of the plasmonic field distribution in order to consistently quantify the mode confinement.

Out of many definitions found in the literature we will provide three different definitions and compare them in our study of different waveguide structures.

The first definition A ₁ depends on the peak energy density and defined as [34],

A ₁ = 1

Max {W (r)}

Z

A

W (r)dA, (3.1)

Here, W (r) is the energy density,

W (r) = 1

2 Re { d[ω(r)]

dω }|E(r)| ² + 1

2 µ 0 |H(r)| ² . (3.2)

A 1 is directly related with the non-linear properties like spontaneous emission rate enhance- ment or Purcell factor. Since A 1 depends on the maximum energy density, this measure may be misleading for waveguide with very sharp features. This is useful to quantify the local field enhancement which is not necessarily accompanied by strong confinement of total energy [34].

The second definition A 2 is a statistical measure requiring integration of energy density over the cross section,

A ₂ =

R

A∞ W (r)dA  2

R

A∞ W (r) ² dA (3.3)

A 2 is usually a better measure since it takes into consideration of the overall field. This

measure has firm foundation in optical fiber theory [35].However, it is potentially sensitive to

energy distribution.

25

The third definition A 3 is made, aiming to gauge confinement irrespective of field distribution.

A ₃ is defined as the minimum area where exactly a portion η, of the mode’s total energy resides. Since η is a generic constant, A ₃ is promised to be a geometry-independent measure of confinement. We choose η = 0.5 in our study. So it will be the minimum area where half of the mode’s total energy will reside. In order to calculate A ₃ , we need to solve the following minimization problem [34],

A 3 = min _{f (r)} Z

A∞

f (r)dA, so that,

Z

A∞

[f (r) − η]W (r)dA = 0

(3.4)

We can iteratively solve the problem by using,

f (r) = 0, if w(r) < W ₀ ,

f (r) = 1, if w(r) > W ₀ , (3.5)

where W ₀ is the contour containing η(= 0.5) of the mode’s total energy. We wrote a Matlab script in order to do this calculation which can be found in the Appendix. This program is capable of calculating A 3 after receiving energy density data from COMSOL.

3.1.2 Propagation length

Although plasmonic structures provide extremely localized electromagnetic fields there is an intrinsic cost involved in terms of the distance that the field can travel because of the damping inside the metal. If the plasmonic mode propagate harmonically in z direction with field variation, exp[i(βz − ωt)], the propagation distance is defined as the distance space that a mode can travel before the energy density decaying to 1/e of its original value [34],

L p = 1

2Im {β} , (3.6)

where, β is the propagation constant defined as β = n ef f k 0 and k 0 is the wave vector in vacuum equals to ^2π _λ

0

. This definition is well established and consistent thorough out the literature.

3.1.3 Figure of merit

Another inconsistency arises when we try to evaluate the performance of a particular waveg-

uide structure. Although most of the plasmonic waveguides works based on the same plas-

monic phenomena, they exhibit different characteristics in terms of propagation, confinement

and the trade-off between them. Figure of merit of a certain structure should take into con-

sideration all these effects. Since the propagation distance is well defined we need to choose

a proper definition of confinement in order to quantify figure of merit objectively. We choose

A 2 , since this is quite consistent and provides almost similar behaviour of the geometry inde- pendent definition A ₃ . A ₂ usually provides the highest value among the three definitions, so we will be using it mostly in order to avoid the probability of overestimation of confinement during the calculation of figure of merit and to avoid the complexity involved in calculating A ₃ . Nevertheless, as we argued A ₃ should be the most consistent definition of mode area, we will try to use all these definition when we look for effective mode area and compare whenever possible.

Having said those, figure of merit is defined as ratio between the normalized propagation length and normalized mode area,

F OM = L p /λ 0

A 2 /A 0

(3.7)

Here, A 0 = ( ^λ ₂

) ² is the diffraction limited mode area and λ 0 is the vacuum wavelength.

3.2 Slot waveguides

This is one of the earlier structure [29] but still provide reasonable performance compare to other plasmonic based waveguides. It is based on a slot in a thin metal film embedded in dielectric as shown in Figure 3.1. The size of the bound mode supported by this waveguide depends on the near field of the slot. The size of confined mode can be in the deep subwave- length range and propagation length is around tens of micrometers at telecommunication wavelength 1.55µm. Such properties are particularly useful for interconnects in hybrid opto- electronic circuits and devices. We will investigate the physics behind their characteristics by comparing to some simplified geometries.

3.2.1 Simplified geometries

The behaviour of the plasmonic slot waveguide that is shown in Figure 3.1 can be understood with the help of some simplified geometries [30]. The corresponding simplified structures includes the film SPP mode of a metal film with dielectric surrounding (DMD, Fig. 3.2(a)), gap SPP mode of a dielectric gap with metal surrounding (MDM, Fig. 3.2(b)) and edge mode of truncated metal film (Fig. 3.2(c)). We will concentrate on the modes in these geometries which are directly related to the fundamental modes in the slot and later we will show that many properties of the 3D slot waveguide asymptotically approach those of the simplified geometries.

We used silver ( m = −129 + 3.3i) as metal and silica ( d = 2.25) as dielectric [15]. Optical

communication wavelength (λ 0 = 1.55µm) has been considered. In Figure 3.2(a) the depen-

dence of effective index on the film thickness h, for the DMD (dielectric-metal-dielectric) is

shown. Although these type of film structures have two modes - higher index short range

27

Figure 3.1: Domain and geometry of a 3D two conductor slot waveguide along with 2D cross-sectional view.

(SR-SPP) and lower index long range (LR-SPP). We are concerned with SR-SPP mode, since the relevance with the highly localized mode of slot structure. As we have seen in the previous chapter when h → 0 , k SR−SP P increases indefinitely which is quite justified in Figure 3.2(a).

On the other hand when h → ∞, the coupling between the surface modes of two interfaces weakens and therefore n _{ef f} approaches n SP P of single interface.

For MDM structure in Figure 3.2(b) has the same dielectric gap as the width of the slot. We will consider the Gap-SPP (G-SPP) for which the longitudinal component has odd symmetry.

We recall our concept of G-SPP in the limit of very narrow gap k G−SP P ≈= − ^2 _

. The field squeezes into the metal when the gap w decreases and therefore both n _{ef f} and loss due to damping increases. Large w correspond to uncoupled SPP, as in the DMD structure.

Figure 3.2(c) and 3.2(d) show the effective index of a truncated metal film as a function of h and power density (poynting vector) profile respectively. The edge mode has maximum power density at the corner of the film because of singular behaviour of electric field near sharp edges [12]. The only difference with DMD structure in terms of dependence with h is that, when h → ∞, n ef f approaches the effective index of the mode of a single 90 ⁰ corner which is little higher than the index of single interface SPP mode.

3.2.2 Characteristic of symmetric slot waveguide

Now we have discussed the simplified structure, we will consider a symmetric plasmonic slot

waveguide structure with slot between two narrow metallic film (silver) which are embedded in

an infinite homogeneous dielectric (silica). We would like to study the dispersion characteristic

of this structure. In order to do that we need to know the complex permittivity of silver at

different wavelength. This is defined by Drude model [12] as,

(a) (b)

(c) (d)

Figure 3.2: (a) Effective index of DMD structure as a function of the thickness of the metal film, h (solid line),(b) Effective refractive index n ef f of a silver-silica-silver or MDM structure as a function of the width of gap region w, refractive index of silica (broken line), (c) n ef f of the fundamental edge mode (truncated Ag film) as a function of thickness of the Ag film (solid line), index of silica (broken line) (d) Power density distribution of truncated edge mode for a Ag film of thickness h = 50nm at λ 0 = 1.55µm.

 _m = 1 − ω ² _p

ω ² + iωγ , (3.8)

where, the plasma frequency ω _p = 1.38 × 10 ¹⁶ rad/s and the collision frequency γ = 3.2258 × 10 ¹³ rad/s for silver (Ag) [15]. The plot of this equation is shown in Figure 3.3(a). Both the real and imaginary part decreases with increase of frequency.

Since we are interested in the regime of sub-wavelength our reference structure has a slot

of width w = 50nm and thickness of silver film h = 50nm embedded in homogeneous silica

( _d = 2.25). The dispersion relation of the fundamental mode is shown in Figure 3.4(a). We

can see that the wave vector k is a bound mode and is larger than all the radiation mode

in silica. We can also show that k of this wave guide is larger than all the metal film SPP

29

(a) (b)

Figure 3.3: (a) Relative permittivity of silver as a function of frequency using the Drude model. (b) Ratio of imaginary and real part of the permittivity of silver as function of frequency.

propagating modes over the entire frequency range.

(a) (b)

Figure 3.4: (a) Dispersion relation of the fundamental mode when w, h = 50nm (b) Propagation length as a function of wavelength for the same structure.

In Figure 3.4(b) we can find the propagation length as a function of wavelength. We recall the definition of propagation length, L p = _2Im[β] ¹ = the distance covered by a mode before the field energy is attenuated by 1/e. In order to understand the dependence of L p on wavelength, we can refer to Figure 3.3(b) where propagation loss (α _{SP P} ∝ Im( m )/Re( _m ) ² ) with respect to frequency is shown. When the wavelength decreases the fraction of modal power in the metal increases, so the propagation length decreases because of increase in damping.

The power density profiles at various wavelength can be found in Figure 3.5(a)-(d). We

observe a highly confined fundamental mode over a wide frequency range. Although at shorter

(a) (b)

(c) (d)

Figure 3.5: Power density profile of the fundamental mode when w, h = 50nm for different wave- length, (a) λ 0 = 0.6µm, (b)λ 0 = 0.6µm,(c) λ 0 = 1µm (telecommunication wavelength), (d) λ 0 = 10µm.

wavelength we have small value of L p , we can achieve the most confined mode as shown in Figure 3.5(a). Since fraction of modal power in metal is higher at shorter wavelength, the fundamental mode is composed mainly of the edge modes which are weakly coupled because of small decay length into the dielectric. As λ increase and energy density at the middle of the slot becomes significant and we are getting close to the light line of silica but still as we can see in Figure 3.5(d) the modal size is far smaller than the wavelength.

Now, we change different geometrical parameters e.g. the width w and the thickness h and investigate the effect of these parameters on modal characteristics. In Figure 3.6(a) we see the real part of the effective index as the gap w varies for fixed h = 50nm at 1.55µm wavelength.

Since the thickness of the film is fixed, for small w, n _{ef f} is very close to that of the MDM structures. As the gap increases n _{ef f} approaches the similar behaviour of the edge mode (shown in dash-dotted line). The propagation length L p increases with the gap size as shown in Figure 3.6(b). This is quite straight forward because most of the power will be in dielectric region for large w.

Figure 3.6(c) shows the behaviour of the effective mode area (as defined in section 3.1)as a

31

(a) (b)

(c)

Figure 3.6: (a) Effective index n ef f as function of slot width w for fixed film thickness h = 50nm

(solid line), the effective indices of the corresponding modes of the simplified structure

are also shown, (b) Dependence of propagation length L p on the width w, (c) Different

effective mode area as a function of w.

function of w according to different definitions. For very small w, the modal character is close to MDM, there is only small effect of edge mode. Because of this A ₁ is little higher than A ₃ since A ₁ is inversely proportional to the maximum energy density. A ₂ , the statistical definition usually provides the highest value among the three. Now as w increases the effect of the edge mode is increasing, so A ₁ will become smaller than A ₃ at some point. Nevertheless, all the three definition follow the same trend, modal area increases with the gap size.

(a)

0 50 100 150

4 6 8 10 12 14 16 18 20

h (nm) Propagationlength,Lp(µm)

w = 50nm λ0= 1.55µm

(b)

(c)

Figure 3.7: (a) Effective index as a function of film thickness h for fixed slot width, w = 50nm, indices of the corresponding MDM and DMD mode are also shown in dotted line, (b) Propagation length as function of h, and (c) Effective modal area (A 1,2,3 as a function of h.

In Figure 3.7(a), (b) and (c) we show the behaviour of effective index n _{ef f} , propagation length

L p and modal areas (A 1,2,3 ) respectively as a function of film thickness h for a fixed gap size